A Short Guide to Celestial Navigation Copyright © 1997-2015 Henning Umland Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Revised June 5th, 2015 First Published May 20th, 1997 .ndeB 1reface Chapter 1he Basics of Celestial ,aEigation Chapter 2 Altitude Measurement Chapter 3 )eographic .osition and 1ime Chapter 4 Finding One's .osition 0ight Reduction) Chapter 5 Finding the .osition of an Advancing 2essel Determination of Latitude and Longitude, Direct Calculation of Chapter 6 .osition Chapter 7 Finding 1ime and Longitude by Lunar Distances Chapter 8 Rise, 0et, 1wilight Chapter 9 )eodetic Aspects of Celestial ,aEigation Chapter 0 0pherical 1rigonometry Chapter 1he ,aEigational 1riangle Chapter 12 )eneral Formulas for ,aEigation Chapter 13 Charts and .lotting 0heets Chapter 14 Magnetic Declination Chapter 15 Ephemerides of the 0un Chapter 16 ,aEigational Errors Chapter 17 1he Marine Chronometer AppendiB -02 ,ree Documentation /icense Much is due to those who first broke the way to knowledge, and left only to their successors the task of smoothing it. Samuel Johnson Preface Why should anybody still practice celestial navigation in the era of electronics and GPS? One might as well ask why some photographers still develop black-and-white photos in their darkroom instead of using a digital camera. The answer would be the same: because it is a noble art, and because it is rewarding. No doubt, a GPS navigator is a powerful tool, but using it becomes routine very soon. In contrast, celestial navigation is an intellectual challenge. Finding your geographic position by means of astronomical observations requires knowledge, skillfulness, and critical judgement. In other words, you have to use your brains. Everyone who ever reduced a sight knows the thrill I am talking about. The way is the goal. It took centuries and generations of navigators, astronomers, geographers, mathematicians, and instrument makers to develop the art and science of celestial navigation to its present level, and the knowledge thus accumulated is a treasure that should be preserved. Moreover, celestial navigation gives us an insight into scientific thinking and creativeness in the pre-electronic age. Last but not least, celestial navigation may be a highly appreciated alternative if a GPS receiver happens to fail. When I read my first book on navigation many years ago, the chapter on celestial navigation with its fascinating diagrams and formulas immediately caught my particular interest although I was a little intimidated by its complexity at first. As I became more advanced, I realized that celestial navigation is not nearly as difficult as it seems to be at first glance. Studying the literature, I found that many books, although packed with information, are more confusing than enlightening, probably because most of them have been written by experts and for experts. On the other hand, many publications written for beginners are designed like cookbooks, i. e., they contain step-by-step instructions but avoid much of the theory. In my opinion, one can not really comprehend celestial navigation and enjoy the beauty of it without knowing the mathematical background. Since nothing really complied with my needs, I decided to write a compact manual for my personal use which had to include the most important definitions, formulas, diagrams, and procedures. As time went by, the project gained its own momentum, the book grew in size, and I started wondering if it might not be of interest to others as well. I contacted a few scientific publishing houses, but they informed me politely that they considered my work as dispensable (“Who is going to read this!”). I had forgotten that scientific publishing houses are run by marketing people, not by scientists. Around the same time, I became interested in the internet, and I quickly found that it is the ideal medium to share one's knowledge with others. Consequently, I set up my own web site to present my book to the public. The style of my work may differ from other books on this subject. This is probably due to my different perspective. When I started the project, I was a newcomer to the world of navigation, but I had a background in natural sciences and in scientific writing. From the very beginning, it has been my goal to provide accurate information in a structured and comprehensible form. The reader may judge whether this attempt has been successful. More people than I expected are interested in celestial navigation, and I would like to thank my readers for their encouraging comments and suggestions. However, due to the increasing volume of correspondence, I am no longer able to answer individual questions or to provide individual support. Unfortunately, I have still a few other things to do, e. g., working for a living. Nonetheless, I keep working on this publication at leisure, and I am still grateful for suggestions and error reports. This publication is released under the terms and conditions of the GNU Free Documentation License. A copy of the latter is included. June 5th, 2015 Henning Umland Web site: http://www.celnav.de Chapter 1 The Basics of Celestial 1avigation Celestial navigation , also called astronomical navigation , is the art and science of finding one s geographic position through astronomical observations, in most cases by measuring altitudes of celestial bodies \ sun, moon, planets, or stars. 2n observer watching the night sky without knowing anything about geography and astronomy might spontaneously get the impression of being on a horizontal plane located at the center of a huge hollow sphere with the celestial bodies attached to its inner surface. This naive concept of a spherical universe has probably been in existence since the beginning of mankind. 9ater, astronomers of the ancient world (Ptolemy et al.) developed it to a high degree of perfection. Still today, spherical astronomy is fundamental to celestial navigation since the navigator, like the astronomers of old, measures apparent positions of bodies in the sky without knowing their absolute positions in space. The apparent position of a body in the sky is best characterized by the horizon system of coordinates which is a special case of a spherical coordinate system . In this system, an imaginary (!) observer is located at the center of the celestial sphere , a hollow sphere of infinite diameter, which is divided into two hemispheres by the plane of the celestial horizon (Fig. 1-1 ). The center of the celestial sphere coincides with the center of the earth which is also assumed to be a sphere. The first coordinate of the observed body is its geocentric altitude , /. 7 is the vertical angle between the celestial horizon and a straight line extending from the center of the celestial sphere to the body. 7 is measured from 0° through +90° above the horizon and from 0° through -90° below the horizon. The geocentric zenith distance , z, is the corresponding angular distance between the body and the zenith , an imaginary point vertically overhead. The zenith distance is measured from 0° through 180°. 7 and z are complementary angles (7 + z 1 90°). The point opposite to the zenith on the celestial sphere is called nadir (7 1 -90°, z 1 180°). 7 and z are also arcs of the vertical circle going through zenith, nadir, and the observed body. The second coordinate of the body, the geocentric true azimuth , Az 1, is the horizontal direction of the body with respect to the geographic north point on the celestial horizon, measured clockwise from 0°(N) through 360°. The third coordinate, the distance of the body from the center of the celestial sphere, is not measured. In reality, the observer is not located on the plane of the celestial horizon but on or above the surface of the earth. The horizontal plane passing through the observer s eye is called sensible horizon ( Fig. 1-2) . The latter merges into the geoidal horizon , a plane tangent to the earth at the observer s position, when the observer s eye is at sea level. The planes of celestial, geoidal, and sensible horizon are parallel to each other and perpendicular to the direction of gravity at the observer s position. 1-1 Since sensible and geoidal horizon are relatively close to each other (compared with the radius of the earth), they can be considered as identical under most practical conditions. None of the above fictitious horizons coincides with the visible horizon , the line where the earth s surface and the sky appear to meet. Usually, the trigonometric calculations of celestial navigation are based on the geocentric altitudes (or geocentric zenith distances) of bodies. Since it is not possible to measure the geocentric altitude of a body directly, it has to be derived from its altitude with respect to the visible or sensible horizon (altitude corrections, chapter 2). The altitude of a body with respect to the visible sea horizon is usually measured with a marine seItant . Measuring altitudes with respect to the (invisible) sensible horizon requires an instrument with an artificial horizon , e. g., a theodolite (chapter 2). 2n artificial horizon is a device that indicates a plane perpendicular to the local direction of gravity, for example by means of a pendulum or a spirit level. 6eocentric altitude and zenith distance of a celestial body are determined by the distance between the terrestrial observer and the geographic position of the body , .2 . 6P is the point where a straight line extending from the center of the earth, C, to the celestial body intersects the earth s surface ( Fig.